On May 30, 1:15 pm, Zuhair <zaljo...@gmail.com> wrote:> On May 29, 5:29 am, Nam Nguyen <namducngu...@shaw.ca> wrote:>>>>>>>> > On 28/05/2013 6:06 AM, Zuhair wrote:>> > > On May 28, 7:44 am, Nam Nguyen <namducngu...@shaw.ca> wrote:> > >> On 26/05/2013 10:17 PM, zuhair wrote:>> > >>> On May 26, 4:49 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:> > >>>> On 26/05/2013 3:52 AM, Zuhair wrote:>> > >>>>> On May 26, 11:03 am, Nam Nguyen <namducngu...@shaw.ca> wrote:> > >>>>>> On 26/05/2013 12:52 AM, Zuhair wrote:>> > >>>>>>> Frege wanted to reduce mathematics to Logic by extending predicates by> > >>>>>>> objects in a general manner (i.e. every predicate has an object> > >>>>>>> extending it).>> > >>>>>> [...]>> > >>>>>>> Now the above process will recursively form typed formulas, and typed> > >>>>>>> predicates.>> > >>>>>> Note your "process" and "recursively".>> > >>>>>>> As if we are playing MUSIC with formulas.>> > >>>>>>> Now we stipulate the extensional formation rule:>> > >>>>>>> If Pi is a typed predicate symbol then ePi is a term.>> > >>>>>>> The idea behind extensions is to code formulas into objects and thus> > >>>>>>> reduce the predicate hierarchy into an almost dichotomous one, that of> > >>>>>>> objects and predicates holding of objects, thus enabling Rule 6.>> > >>>>>>> What makes matters enjoying is that the above is a purely logically> > >>>>>>> motivated theory, I don't see any clear mathematical concepts involved> > >>>>>>> here, we are simply forming formulas in a stepwise manner and even the> > >>>>>>> extensional motivation is to ease handling of those formulas.> > >>>>>>> A purely logical talk.>> > >>>>>> Not so. "Recursive process" is a non-logical concept.>> > >>>>>> Certainly far from being "a purely logical talk".>> > >>>>> Recursion is applied in first order logic formation of formulas,>> > >>>> Such application isn't purely logical. Finiteness might be a purely> > >>>> logical concept but recursion isn't: it requires a _non-logical_> > >>>> concept (that of the natural numbers).>> > >>>>> and all agrees that first order logic is about logic,>> > >>>> That doesn't mean much and is an obscured way to differentiate between> > >>>> what is of "purely logical" to what isn't.>> > >>> Yes I do agree that this way is not a principled way of demarcating> > >>> logic. I generally tend to think that logic is necessary for analytic> > >>> reasoning, i.e. a group of rules that make possible to have an> > >>> analytic reasoning. Analytic reasoning refers to inferences made with> > >>> the least possible respect to content of statements in which they are> > >>> carried, thereby rendering them empirically free. However this is too> > >>> deep. Here what I was speaking about do not fall into that kind of> > >>> demarcation, so it is vague as you said. I start with something that> > >>> is fairly acceptable as being "LOGIC", I accept first order logic> > >>> (including recursive machinery forming it) as logic, and then I expand> > >>> it by concepts that are very similar to the kind of concepts that made> > >>> it, for example here in the above system you only see rules of> > >>> formation of formulas derived by concepts of constants, variables,> > >>> quantifying, definition, logical connectivity and equivalents,> > >>> restriction of predicates. All those are definitely logical concepts,> > >>> however what is added is 'extension' which is motivated here by> > >>> reduction of the object/predicate/predicate hierarchy, which is a> > >>> purely logical motivation, and also extensions by the axiom stated> > >>> would only be a copy of logic with identity, so they are so innocuous> > >>> as to be considered non logical.> > >>> That's why I'm content with that sort of definitional extensional> > >>> second order logic as being LOGIC. I can't say the same of Z, or ZF,> > >>> or the alike since axioms of those do utilize ideas about structures> > >>> present in mathematics, so they are mathematically motivated no doubt.> > >>> NF seems to be logically motivated but it use a lot of mathematics to> > >>> reach that, also acyclic comprehension uses graphs which is a> > >>> mathematical concept. But here the system is very very close to logic> > >>> that I virtually cannot say it is non logical. Seeing that second> > >>> order arithmetic is interpretable in it is a nice result, it does> > >>> impart some flavor of logicism to traditional mathematics, and> > >>> possibly motivates logicism for whole of mathematics. Mathematics> > >>> might after all be just a kind of Symbolic Logic as Russell said.>> > >>> Zuhair>> > >>>>> similarly here> > >>>>> although recursion is used yet still we are speaking about logic,> > >>>>> formation of formulas in the above manner is purely logically> > >>>>> motivated.>> > >>>> "Purely logically motivated" isn't the same as "purely logical".>> > >>> A part from recursion, where is the mathematical concept that you> > >>> isolate with this system?>> > >> I don't remember what you'd mean by "this system", but my point would be> > >> the following.>> > >> In FOL as a framework of reasoning, any form of infinity (induction,> > >> recursion, infinity) should be considered as _non-logical_ .>> > >> The reason is quite simple: in the language L of FOL (i.e. there's no> > >> non-logical symbol), one can not express infinity: one can express> > >> "All", "There exists one" but one simply can't express infinity.>> > >> Hence _infinity must necessarily be a non-logical concept_ . Hence the> > >> concept such the "natural numbers" can not be part of logical reasoning> > >> as Godel and others after him have _wrongly believed_ .>> > >> Because if we do accept infinity as part of a logical reasoning,> > >> we may as well accept _infinite formulas_ and in such case it'd> > >> no longer be a human kind of reasoning.>> > > I see, you maintain the known prejudice that the infinite is non> > > logical? hmmm... anyhow this is just an unbacked statement.>> > I did; you just don't recognize it apparently: my "The reason is quite> > simple:" paragraph.>> > > I don't see any problem between infinity and logic,>> > Well, then, why don't you express infinity with purely logical> > symbols, for us all in the 2 fora to see? Seriously, that would> > be a great achievement!>> Infinity: Exist x (0 E x & (for all y. y E x -> {y} E x))>> where E is defined as in the head post.>> while 0 and {y} are defined as:>> 0=e(contradictory)> {y}=e{isy}

a typocorrection: {y}=e(isy)>> Where 'contradictory' is defined as: for all x. contradictory(x) iff> ~x=x> and 'isy' is defined as: for all x. isy(x) iff x=y>> I consider the monadic symbol "e" as a "logical" symbol, also identity> symbol is logical.>> The above infinity is a theorem of this logic.>> Zuhair